# Expectation value of multilinear forms over independent Gaussian vectors

Let $$A$$ be a symmetric multilinear form on $$\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$$ and consider the random variable: \begin{align*} X=A(g_1,\ldots,g_n,g_1,\ldots,g_n) \end{align*} where $$g_i\in \mathbb{R}^d$$ are independent random vectors with i.i.d. random Gaussian entries of mean $$0$$ and variance $$1$$. Is there a closed form expression for $$\mathbb{E}(X^2)$$? In the case $$n=1$$ I know that $$\mathbb{E}(X^2)=\|A\|_{F}^2$$, where $$\|\cdot\|_{F}$$ is the Frobenius norm of the underlying norm. Does that generalize in any way for $$n>1$$? Thanks for any help!

For $$n=1$$, $$EX^2=(\text{tr}\,A)^2+2\|A\|_F^2, \tag{1}$$ not $$\|A\|_F^2$$.
Indeed, writing $$A=(a_{ij})$$ and $$g_1=(Z_1,\dots,Z_d)$$ with iid standard normal $$Z_i$$'s, we have $$X=\sum_{i,j}a_{ij}Z_iZ_j$$ and hence $$EX^2=\sum_{i,j,k,l}a_{ij}a_{kl}\,EZ_iZ_jZ_kZ_l.$$ Next, $$EZ_iZ_jZ_kZ_l\ne0$$ only if (i) $$i=j\ne k=l$$ (in which case $$EZ_iZ_jZ_kZ_l=1$$) or (ii) $$i=k\ne j=l$$ (in which case $$EZ_iZ_jZ_kZ_l=1$$) or (iii) $$i=l\ne j=k$$ (in which case $$EZ_iZ_jZ_kZ_l=1$$) or (iv) $$i=j=k=l$$ (in which case $$EZ_iZ_jZ_kZ_l=3$$). So, taking the symmetry of $$A$$ into account, we have $$EX^2=\sum_{i\ne k}a_{ii}a_{kk}+\sum_{i\ne j}a_{ij}^2+\sum_{i\ne j}a_{ij}a_{ji}+3\sum_i a_{ii}^2 \\ =\Big(\sum_i a_{ii}\Big)^2+2\sum_{i,j}a_{ij}^2,$$ so that (1) follows.
For $$n>1$$, $$EX^2$$ will be a big sum over all partitions of the set $$\{1,\dots,4n\}$$ into subsets of even cardinalities (cf. cases (i)--(iv) above).